Standardised Scores
Learn Standardised Scores for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. A standardised score measures how many standard deviations a value is from the mean, allowing fair comparison.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
A standardised score expresses how many standard deviations a value lies from the mean, allowing values from different distributions to be compared fairly. This is a Higher-only topic.
The formula is \(z = \frac{x - \bar{x}}{\sigma}\), where \(x\) is the value, \(\bar{x}\) the mean and \(\sigma\) the standard deviation.
A positive score means the value is above the mean and a negative score below it. A score of \(1.5\) means the value is one and a half standard deviations above the mean. This is how marks from two exams of different difficulty can be compared: the higher standardised score represents the better relative performance, even if the raw mark is lower.
Revision notes
The formula
\(z = \frac{x - \bar{x}}{\sigma}\), where \(x\) is the value, \(\bar{x}\) the mean and \(\sigma\) the standard deviation.
Subtract the mean from the value, then divide by the standard deviation. Keep the sign — it tells you which side of the mean the value lies.
Interpreting the score
Positive: the value is above the mean. Negative: below the mean.
The size tells you how many standard deviations away it is. A score of 1.5 means one and a half standard deviations above the mean; a score of 0 means exactly at the mean.
Comparing across distributions
Two exams of different difficulty produce marks that cannot be compared directly.
Standardising both converts them to a common scale. The higher standardised score represents the better relative performance, even if the raw mark is lower.
Key points
- \(z = \frac{x - \bar{x}}{\sigma}\).
- A positive score is above the mean.
- A negative score is below the mean.
- The size gives the number of standard deviations.
- Standardising allows fair comparison.
- The higher score is the better relative performance.
Worked examples
Example 1
A mark of 68 comes from a distribution with mean 60 and standard deviation 5. Work out the standardised score. [2 marks]
Working
Example 2
A student scores 55 in Test A (mean 50, sd 4) and 70 in Test B (mean 68, sd 8). State which was the better performance. [3 marks]
Working
Example 3
A standardised score is \(-0.8\). Interpret this. [2 marks]
Working
Common mistakes
Losing the negative sign.
It shows the value lies below the mean and must be kept.
Dividing by the mean instead of the standard deviation.
The denominator is the standard deviation.
Comparing raw marks from different tests.
Standardise both before comparing.
Assuming the higher raw mark is the better performance.
The higher standardised score is what matters.
Exam tips
- Subtract the mean before dividing.
- Keep the sign of your answer.
- Standardise both values before comparing.
- Remember this is a Higher-only topic.
Key terms
- Standardised score
- How many standard deviations a value is from the mean.
- \(z\)
- The symbol for a standardised score.
- Above the mean
- Indicated by a positive standardised score.
- Relative performance
- Performance compared with others in the same distribution.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.